On the tree-likeness of hyperbolic spaces

TL;DR

This paper constructs rooted R-trees within proper hyperbolic geodetic spaces, capturing their boundary structure and quasi-geodetic rays, thus revealing the tree-like nature of hyperbolic spaces.

Contribution

It introduces a method to embed hyperbolic spaces into R-trees with boundary properties linked to the space's Assouad dimension.

Findings

Rooted R-trees are constructed with quasi-geodetic rays.

The boundary of the R-tree reflects the hyperbolic boundary structure.

The number of disjoint rays to a boundary point is bounded by the Assouad dimension.

Abstract

In proper hyperbolic geodetic spaces we construct rooted $\mathbb R$-trees with the following properties. On the one hand, every ray starting at the root is quasi-geodetic; so these $\mathbb R$-trees represent the space itself well. At the same time, the trees boundary reflects the boundary of the space in that the number of disjoint rays to a boundary point is bounded in terms of the (Assouad) dimension of the hyperbolic boundary.